Problem: Simplify. Rewrite the expression in the form $10^n$. $\left(10^1\right)^{3}=$
Solution: $\begin{aligned} \left(10^1\right)^{3}&=10^{1\cdot 3} \\\\ &=10^{3} \end{aligned}$ This follows from the general rule $\left(x^m\right)^{n}=x^{m\cdot n}$. We can also see this is correct by expanding the powers. $\begin{aligned} \left(10^1\right)^{3}&=\underbrace{10^1\cdot 10^1\cdot 10^1}_\text{3 times} \\\\\\ &=\underbrace{ \underbrace{10}_\text{1 time} \cdot \underbrace{10}_\text{1 time} \cdot \underbrace{10}_\text{1 time}} _\text{3 times} \\\\ &=10^{3} \end{aligned}$ In conclusion, $\left(10^1\right)^{3}=10^{3}$.